SEASON 1 EP 3 -- DIFFERENTIABILITY

 S1 EPISODE 3

DIFFERENTIABILITY

INTRODUCTION:-

Have you ever wondered what makes a function differentiable?

A function is formally considered differentiable if its derivative exists at each point in its domain, but what does this mean?

It means that a function is differentiable everywhere its derivative is defined.

So, as long as you can evaluate the derivative at every point on the curve, the function is differentiable.

Differentiability in Calculus refers to the property of a function that allows it to have a derivative at a given point. In other words, a function is differentiable at a point if its derivative exists at that point. The concept of differentiability is central to calculus and has significant implications in both theoretical and applied mathematics.

Derivative Definition:-

The derivative of a function $f(x)\; at\; a\; point$$x = a$ is defined as the limit:

$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

If this limit exists, the function is said to be differentiable at $x = a$.

                                       

Geometric Interpretation:-

Differentiability means that the graph of the function has a well-defined tangent line at the point $(a, f(a))$. The slope of this tangent is given by $f^{\mathrm{\prime }}(a)$

Differentiability vs. Continuity:-

• Differentiability implies continuity: If $$f(x) is differentiable at $x=\mathrm{a,\; it\; must\; also\; be\; continuous\; at }$$x = a$.
• Continuity does not imply differentiability: A function can be continuous at $x = a$ but not differentiable there (e.g., $f(x) = |x|$ at $x=\mathrm{0).}$

Differentiability Over an Interval:-

A function $f(x)$ is differentiable on an interval $(a,b)$ if it is differentiable at every point in $(a,b)$. If it is differentiable on a closed interval $[a, b]$, the derivatives at the endpoints are defined as one-sided limits.

EXAMPLES:-

1. Polynomial Function:

A polynomial $f(x) = x^2$ is differentiable everywhere because it is smooth and continuous with no sharp corners or vertical tangents.

$$f'(x) = 2x$$

2. Absolute Value Function:

The function $f(x) = |x|$ is not differentiable at $x = 0$ because it has a sharp corner at that point.

$$f'(x) = \begin{cases} 1 & \text{if } x > 0, \\ -1 & \text{if } x < 0. \end{cases}$$

3. Trigonometric Function:

The sine function $f(x) = \sin(x)$ is differentiable everywhere with:

$$$$f′(x)=cos⁡(x)

f'(x) = \cos(x)

Applications of Differentiability:-

1. Optimization: Finding maxima and minima using derivatives.
2. Physics: Describing rates of change, such as velocity and acceleration.
3. Economics: Analyzing marginal cost and revenue functions.
4. Engineering: Designing curves and surfaces in CAD systems.

So, here we come to an end of the series, hope it was helpful for you and you gained knowledge ,we will come up with new topic in new series till then stay tuned.

KEEP LEARNING 🕮

KEEP SHINING !!

THANK YOU 😊

CIA 1(A)

NAME- SWARA ARYA 

REG NO. 24215226

CLASS-2BD

REFRENCES:-

• BYJUS
• KHAN ACADEMY
• WIKIPEDIA
• BOOK- Differential calculus for beginners by Joseph Edwards
• GOOGLE